【高数】求幂级数的和函数：Σn=1->无穷 n(n+1)x^n

【高数】求幂级数的和函数：Σn=1->无穷 n(n+1)x^n
我是这样算的
S(x)=Σn=1->无穷 n(n+1)x^n
两边取积分
右边=Σn=1->无穷 nx^(n+1)=x^2*(Σn=0->无穷 nx^(n-1))
令S1(x)=Σn=1->无穷 nx^(n-1)
两边取积分
右边=Σn=1->无穷 x^n=x/(1-x)
故S1(x)=(x/(1-x))'=1/(1-x)^2
S(x)=(x^2*S1(x))'=(x^2/(1-x)^2)'=2x/(1-x)^3
但答案是2/(1-x)^3,我这么多了个x?